191] NORMAL MODES OF A SPHERICAL SHEET. 317 



round the globe, the velocity of propagation, as measured at the 

 equator, being 



* (7). 



It is easily verified, on examination, that the orbits of the 



particles are now ellipses having their principal axes in the 



directions of the meridians and parallels, respectively. At the 

 equator these ellipses reduce to straight lines. 



In the case n=l, the harmonic is always of the zonal type. 

 The harmonic spheroid (4) is then, to our order of approximation, 

 a sphere excentric to the globe. It is important to remark, 

 however, that this case is, strictly speaking, not included in our 

 dynamical investigation, unless we imagine a constraint applied to 

 the globe to keep it at rest; for the deformation in question of 

 the free surface would involve a displacement of the centre of mass 

 of the ocean, and a consequent reaction on the globe. A corrected 

 theory for the case where the globe is free could easily be investi- 

 gated, but the matter is hardly important, first because in such 

 a case as that of the Earth the inertia of the solid globe is so 

 enormous compared with that of the ocean, and secondly because 

 disturbing forces which can give rise to a deformation of the type 

 in question do not as a rule present themselves in nature. It 

 appears, for example, that the first term in the expression for the 

 tide-generating potential of the sun or moon is a spherical har- 

 monic of the second order (see Appendix). 



When n = 2, the free surface at any instant is approximately 

 ellipsoidal. The corresponding period, as found from (5), is then 

 *816 of that belonging to the analogous mode in an equatorial 

 canal (Art. 178). 



For large values of n the distance from one nodal line to 

 another is small compared with the radius of the globe, and the 

 oscillations then take place much as on a plane sheet of water. 

 For example, the velocity, at the equator, of the sectorial waves 

 represented by (6) tends with increasing n to the value (gh)%, in 

 agreement with Art. 167. 



From a comparison of the foregoing investigation with the general theory 

 of Art. 165 we are led to infer, on physical grounds alone, the possibility of 

 the expansion of any arbitrary value of in a series of surface harmonics, thus 



